×

zbMATH — the first resource for mathematics

Nikodým convergence theorem for uniform space valued functions defined on \(D\)-posets. (English) Zbl 0856.28008
The paper deals with a generalization of the classical Nikodým theorem stating that the limit of a sequence of countable additive measures is again a countable additive measure. The authors deal with functions defined on so-called \(D\)-posets [see F. Kôpka and F. Chovanec, Math. Slovaca 44, No. 1, 21-34 (1994; Zbl 0789.03048)] with values in a uniform space. Instead of countable additivity so-called exhaustiveness is considered.

MSC:
28C99 Set functions and measures on spaces with additional structure
28E10 Fuzzy measure theory
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] BIRKHOFF G.: Lattice Theory. (3rd edition). Amer. Math. Soc. Colloq. Publ. 25, Amer. Math. Soc, Providence, RI, 1967. · Zbl 0153.02501
[2] CHANG C. C.: Algebraic analysis of many valued logics. Trans. Amer. Math. Soc. 88 (1958), 467-490. · Zbl 0084.00704
[3] CONSTANTINESCU C.: Spaces of Measures. Walter de Gruyter, Berlin-New York, 1984. · Zbl 0555.28004
[4] CONSTANTINESCU C.: Nikodým boundedness theorem. Libertas Math. 1 (1981), 51-73. · Zbl 0482.28009
[5] COOK T. A.: The Nikodým-Hahn-Vitali-Saks theorem for states in a quantum logic. Mathematical Foundations of Quantum Theory, Academic Press, London, 1978, pp. 275-285.
[6] de LUCIA P., DVUREČENSKIJ A.: Decompositions of Riesz space-valued measures on orthomodular posets. Tatra Mountains Math. Publ. 2 (1993), 229-239. · Zbl 0803.28006
[7] de LUCIA P., DVUREČENSKIJ A.: Yosida-Hewitt decompositions of Riesz space-valued measures on orthoalgebras. Tatra Mountains Math. Publ. 3 (1993), 101-110. · Zbl 0804.28005
[8] de LUCIA P., MORALES P.: Non-commutative decomposition theorems in Riesz spaces. Proc. Amer. Math. Soc. 120 (1994), 193-202. · Zbl 0796.28008
[9] DVUREČENSKIJ A.: On convergences of signed states. Math. Slovaca 28 (1978), 289-295. · Zbl 0421.28003
[10] DVUREČENSKIJ A.: Regular measures and completeness of inner product spaces. Contrib. General Algebras 7, Holder-Pichler-Tempski; Verlag B. G. Teubner, Wien; Stuttgart, 1991, pp. 137-147. · Zbl 0795.28011
[11] DVUREČENSKIJ A.: Completeness of inner product spaces and quantum logic of splitting subspaces. Lett. Math. Phys. 15 (1988), 231-235. · Zbl 0652.46017
[12] DVUREČENSKIJ A.: Gleason’s Theorem and Applications. Kluwer Academic Publ.; Ister Science Press, Dordrecht-Boston-London; Bratislava, 1993. · Zbl 0795.46045
[13] DVUREČENSKIJ A., RIEČAN B.: Fuzzy quantum models. Internat. J. General Systems 20 (1991), 39-54. · Zbl 0746.60004
[14] DVUREČENSKIJ A., RIEČAN B.: Decompositions of measures on orthoalgebras and difference posets. Internat. J. Theoret. Phys. 33 (1994), 1387-1402. · Zbl 0815.03038
[15] FOULIS D. J., GREECHIE R. J., RUTTIMANN G. T.: Filters and supports in orthoalgebras. Internat. J. Theoret. Phys. 31 (1992), 787-807. · Zbl 0764.03026
[16] KALMBACH G.: Orthomodular Lattices. Acad. Press, London-New York, 1983. · Zbl 0528.06012
[17] KLEMENT E. P., WEBER S.: Generalized measures. Fuzzy Sets and Systems 40 (1991), 375-394. · Zbl 0733.28012
[18] KÔPKA F.: D-posets of fuzzy sets. Tatra Mountains Math. Publ. 1 (1992), 83-87. · Zbl 0797.04011
[19] KÔPKA F., CHOVANEC F.: D-posets. Math. Slovaca 44 (1994), 21-34. · Zbl 0789.03048
[20] LUXEMBURG W. A. J., ZAANEN A. C.: Riesz Spaces I. North-Holland, Amsterdam-London, 1971. · Zbl 0231.46014
[21] MUNDICI D.: Interpretation of AFC*-algebras in Lukasiewicz sentential calculus. J. Funct. Anal. 65 (1986), 15-53.
[22] NAVARA M., PTÁK P.: Difference posets and orthoalgebras. · Zbl 0691.03045
[23] PAP E.: Decompositions of supermodular functions and \square -decomposable measures. Fuzzy Sets and Systems 65 (1994), 71-83. · Zbl 0859.28012
[24] PAP E.: On non-additive set functions. Atti. Sem. Mat. Fis. Univ. Modena 39 (1991), 345-360. · Zbl 0732.28009
[25] PAP E.: The Brooks-Jewett theorem for non-additive set functions. Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 21 (1991), 75-82. · Zbl 0753.28011
[26] PTÁK P., PULMANNOVÁ S.: Orthomodular Structures as Quantum Logics. Kluwer Acad. Press, Dordrecht, 1991. · Zbl 0743.03039
[27] RANDALL C., FOULIS D.: New Definitions and Theorems. University of Massachusetts Mimeographed Notes, Amherst, Massachusetts, 1979.
[28] RANDALL C., FOULIS D.: Empirical logic and tensor products. Interpretations and Foundations of quantum Theory. Vol. 5 (H. Neumann, Wissenschaftsverlag, Bibliographisches Institut, Mannheim, 1981, pp. 9-20. · Zbl 0495.03041
[29] RÜTTIMANN G. T.: The approximate Jordan-Hahn decomposition. Canad. J. Math. 41 (1989), 1124-1146. · Zbl 0699.28001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.